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arXiv:2501.06100 · IEEE Access

Practical Quantum Circuit Implementation for Simulating Coupled Classical Oscillators

Natt Luangsirapornchai, Peeranat Sanglaor, Apimuk Sornsaeng, Stéphane Bressan, Thiparat Chotibut, Kamonluk Suksen, Prabhas Chongstitvatana

Published Jan 1, 2025• 3 citations
arXivHTMLPDF

Abstract

Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al. (2023), we present and implement a detailed quantum circuit construction for simulating one-dimensional spring–mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform mass–spring setting, our circuit construction requires a gate complexity of$\mathcal {O}\bigl (\log _{2}^{2} N\,\log _{2}(1/\varepsilon )\bigr )$, where N is the number of oscillators and$\varepsilon $is the target accuracy of the approximation. For more general, heterogeneous mass–spring systems, the total gate complexity is$\mathcal {O}\bigl (N\log _{2} N\,\log _{2}(1/\varepsilon )\bigr )$. Both settings require$\mathcal {O}(\log _{2} N)$qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.

DOI
10.1109/access.2025.3551308

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Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al. (2023), we present and implement a detailed quantum circuit construction for simulating one-dimensional spring–mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform mass–spring setting, our circuit construction requires a gate complexity ofO(log⁡22N log⁡2(1/ε))\mathcal {O}\bigl (\log _{2}^{2} N\,\log _{2}(1/\varepsilon )\bigr )O(log22​Nlog2​(1/ε)), where N is the number of oscillators andε\varepsilon εis the target accuracy of the approximation. For more general, heterogeneous mass–spring systems, the total gate complexity isO(Nlog⁡2N log⁡2(1/ε))\mathcal {O}\bigl (N\log _{2} N\,\log _{2}(1/\varepsilon )\bigr )O(Nlog2​Nlog2​(1/ε)). Both settings requireO(log⁡2N)\mathcal {O}(\log _{2} N)O(log2​N)qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.
We investigate the quantum algorithm of Babbush et al. (arXiv:2303.13012v3) for simulating coupled harmonic oscillators, which promises exponential speedups over classical methods. Focusing on linearly connected oscillator chains, we bridge the gap between theory and implementation by developing and comparing three concrete realizations of the algorithm. First, we implement a sparse initial state preparation combined with product-formula (Suzuki-Trotter) Hamiltonian simulation. Second, we implement a fully quantum, oracle-based framework in which classical data are accessed via oracles, the Hamiltonian is block-encoded, and time evolution is performed using QSVT-based Hamiltonian simulation. Third, we propose an efficient alternative that combines the sparse state-preparation routine of the first approach with the oracle and block-encoding-based simulation pipeline of the second. We provide these implementations on Classiq, a high-level quantum design platform and provide appropriate resource benchmarks. Our simulation results show that the complex initial state preparation proposed by Babbush et al. can be circumvented at least in the linear-chain case. Finally, we illustrate two physical applications-extracting normal modes and simulating coarse-grained energy propagation-demonstrating how the algorithm connects to measurable observables. Our results clarify the resource requirements of the algorithm and provide concrete pathways toward practical quantum advantage.
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We study the problem of simulating the time evolution of a system of 2nclassical coupled oscillators (e.g., 2nballs connected by springs) on a quantum computer. We map Newton’s equation for harmonic potentials to Schrödinger’s equation, such that the amplitudes of an O(n)\mathcal{O}(n)O(n)-qubit quantum state encode the momenta and displacements of the 2nclassical oscillators. Given oracle access to the masses and spring constants, we describe a quantum algorithm with query and time complexity poly (n) that solves this problem when certain parameters are polynomially bounded and the initial state is easy to prepare. As an example application, we apply our quantum algorithm to efficiently estimate the normalized kinetic energy of an oscillator at any time. We then show that any classical algorithm solving the same problem must make 2Ω(n)2^{\Omega(n)}2Ω(n) queries to the oracle and we also show that when the oracles are instantiated by poly (n)-size circuits, the problem is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state ∣ψ⟩|ψ\rangle∣ψ⟩, a Hermitian matrix HHH, a polynomial fff, and a set of observables {O1,…,Om}\{O_1,\dots,O_m\}{O1​,…,Om​}, the goal is to estimate ⟨ψ∣f(H)†Ojf(H)∣ψ⟩\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle⟨ψ∣f(H)†Oj​f(H)∣ψ⟩ for all j∈{1,…,m}j\in\{1,\dots,m\}j∈{1,…,m}. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of f(H)f(H)f(H). It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with HHH and ∣ψ⟩|ψ\rangle∣ψ⟩ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from HHH, fff, and ∣ψ⟩|ψ\rangle∣ψ⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)∣ψ⟩f(H)|ψ\ranglef(H)∣ψ⟩. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
Abstract Hamiltonian simulation is one of the most important problems in quantum computation, and quantum singular value transformation (QSVT) is an efficient way to simulate a general class of Hamiltonians. However, the QSVT circuit typically involves multiple ancilla qubits and multi-qubit control gates. In order to simulate a certain class of n -qubit random Hamiltonians, we propose a drastically simplified quantum circuit that we refer to as the minimal QSVT circuit, which uses only one ancilla qubit and no multi-qubit controlled gates. We formulate a simple metric called the quantum unitary evolution score (QUES), which is a scalable quantum benchmark and can be verified without any need for classical computation. Under the globally depolarized noise model, we demonstrate that QUES is directly related to the circuit fidelity, and the potential classical hardness of an associated quantum circuit sampling problem. Under the same assumption, theoretical analysis suggests there exists an ‘optimal’ simulation time t opt ≈ 4.81, at which even a noisy quantum device may be sufficient to demonstrate the potential classical hardness.
Block Encoding (BE) is a crucial subroutine in many modern quantum algorithms, including those with near-optimal scaling for simulating quantum many-body systems, which often rely on Quantum Signal Processing (QSP). Currently, the primary methods for constructing BEs are the Linear Combination of Unitaries (LCU) and the sparse oracle approach. In this work, we demonstrate that QSP-based techniques, such as Quantum Singular Value Transformation (QSVT) and Quantum Eigenvalue Transformation for Unitary Matrices (QETU), can themselves be efficiently utilized for BE implementation. Specifically, we present several examples of using QSVT and QETU algorithms, along with their combinations, to block encode Hamiltonians for lattice bosons, an essential ingredient in simulations of high-energy physics. We also introduce a straightforward approach to BE based on the exact implementation of Linear Operators Via Exponentiation and LCU (LOVE-LCU). We find that, while using QSVT for BE results in the best asymptotic gate count scaling with the number of qubits per site, LOVE-LCU outperforms all other methods for operators acting on up to ≲ 11 qubits, highlighting the importance of concrete circuit constructions over mere comparisons of asymptotic scalings. Using LOVE-LCU to implement the BE, we simulate the time evolution of single-site and two-site systems in the lattice φ 4 theory using the Generalized QSP algorithm and compare the gate counts to those required for Trotter simulation.